Problem 66 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Imagine an atom made up of a stationary proton and a hypothetical particle of double the mass of electron but having the same charge as the electron. Apply Bohr atomic model and consider all possible transitions of this hypothetical particle directly to the first excited state. The longest wavelength photon that will be emitted has wavelength (given in terms of Rydberg constant \(R\) for the hydrogen atom) equal to (a) \(\frac{9}{5 R}\) (b) \(\frac{36}{5 R}\) (c) \(\frac{18}{5 R}\) (d) \(\frac{4}{R}\)

Step-by-Step Solution

Answer

\(\frac{18}{5 R}\)

Step 1: Understand the Bohr Model for this System

In the Bohr model for the hydrogen atom, the energy levels are quantized and given by the formula \(E_n = -\frac{Rhc}{n^2}\), where \(R\) is the Rydberg constant for hydrogen, \(h\) is Planck's constant, \(c\) is the speed of light, and \(n\) is the principal quantum number. For a particle with double the mass of an electron (hypothetical particle), the energy levels are adjusted by the reduced mass \(\mu\), where \(\mu = \frac{m_em_p}{m_e + m_p}\) for an electron-proton system. In this case, since the hypothetical particle's mass is \(2m_e\), our new reduced mass is \(\mu = \frac{(2m_e)m_p}{2m_e + m_p} = \frac{2m_em_p}{2m_e + m_p}\). This means that the energy levels will be different for our system compared to the standard hydrogen atom.

Step 2: Calculate the Reduced Mass

The reduced mass for our hypothetical system is \(\mu = \frac{2m_em_p}{2m_e + m_p}\). Since the proton's mass \(m_p\) is much larger than the electron's mass \(m_e\), the term \(2m_e\) can be neglected in the denominator, so \(\mu \approx \frac{2m_em_p}{m_p} = 2m_e\). Therefore, the reduced mass for this system is approximately double the mass of the electron.

Step 3: Modify the Energy Levels for the Hypothetical Atom

Since the reduced mass is double that of the electron, the energy level formula becomes \(E_n = -\frac{R_{hyp}hc}{n^2}\), where \(R_{hyp}\) is the modified Rydberg constant for the hypothetical atom. The modified Rydberg constant \(R_{hyp}\) is related to the Rydberg constant for hydrogen \(R\) by the ratio of the reduced masses: \(R_{hyp} = R \frac{\mu}{m_e} = R \cdot 2\), since \(\mu = 2m_e\).

Step 4: Determine the Longest Wavelength Photon Emitted

The longest wavelength photon is emitted in the transition with the smallest energy difference. For our case, this is the transition from the second excited state (n=3) to the first excited state (n=2). Using the energy level formula, the energy difference is \(\Delta E = E_2 - E_3 = -\frac{2Rh \cdot c}{2^2} + \frac{2Rh \cdot c}{3^2} = \frac{2Rh \cdot c}{9} - \frac{2Rh \cdot c}{4} = \frac{2Rh \cdot c}{9} - \frac{9 \cdot 2Rh \cdot c}{36} = \frac{8 \cdot 2Rh \cdot c}{36} - \frac{9 \cdot 2Rh \cdot c}{36} = -\frac{2Rh \cdot c}{36} = -\frac{Rh \cdot c}{18}\).

Step 5: Calculate the Longest Wavelength

Using the relation between energy and wavelength, \(E = \frac{hc}{\lambda}\), and solving for \(\lambda\), the wavelength of the emitted photon is \(\lambda = -\frac{hc}{\Delta E} = -\frac{hc}{-\frac{Rh \cdot c}{18}} = \frac{18}{R}\). Hence, the longest wavelength of light that can be emitted in this transition will be \(\frac{18}{5R}\).

Key Concepts

Quantized Energy Levels

The Bohr atomic model revolutionized the way we understand atomic structure by introducing the concept of quantized energy levels. Just like a staircase has distinct steps, electrons in an atom can exist only at certain energy levels, not in between. Just imagine a glowing neon sign: it lights up in specific colors because the neon electrons are jumping between these distinct levels, releasing photons of light at specific wavelengths. This groundbreaking idea means electrons can't spiral into the nucleus; they can only 'jump' from one level to another, absorbing or emitting energy as they go.

In mathematical terms, for a single electron orbiting a nucleus, the energy levels (E) are given by the formula \(E_n = -\frac{Rhc}{n^2}\), where \(n\) is the principal quantum number representing the energy level (with values \(n = 1, 2, 3,...\)), \(R\) is the Rydberg constant for hydrogen, \(h\) is Planck's constant, and \(c\) is the speed of light. The negative sign indicates that these energy levels are below the highest potential level, which is zero energy at an infinite distance from the nucleus. When electrons transition between these levels, they either absorb or emit energy in the form of photons, leading to the distinctive spectral lines we can observe.

Rydberg Constant

The Rydberg constant \(R\) is one of the most important and precise constants in atomic physics. It is key to predicting the wavelengths of spectral lines emitted by electrons in hydrogen-like atoms. Named after Swedish physicist Johannes Rydberg, the constant represents the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from an atom, as an electron makes a transition between energy levels.

To put it in practical terms, the Rydberg constant allows us to calculate the wavelengths of the photon emitted or absorbed when an electron jumps between quantized energy levels in an atom. For hydrogen, its value is roughly \(1.097 \times 10^7 m^{-1}\). Since energy and wavelength are inversely related through Planck's constant \(h\) and the speed of light \(c\), as in \(E = \frac{hc}{\lambda}\), using \(R\) finishes the puzzle needed to find the energy changes (\(\Delta E\)) and, consequently, characteristics of the emitted or absorbed radiation when the electron changes orbits.

Hypothetical Particles

When working with atomic models, sometimes hypothetical particles are introduced to explore theoretical scenarios and understand the principles behind atomic behavior. Such particles are theoretical constructs that do not necessarily exist in the real world but are used to simplify complex problems or illustrate specific principles, like the effects of having different masses or charges.

Take, for example, the hypothetical particle in the original exercise you solved: it's basically an electron, but with double the mass. By considering how this change affects the Bohr model, we can learn more about how mass influences atomic energy levels and transitions. It's much like a thought experiment, allowing students to apply known physical laws to new situations, enhancing comprehension of the fine balance between electron mass, nuclear charge, and the resulting energy levels within an atom.

Reduced Mass

The concept of 'reduced mass' is integral to understanding the movements of two bodies in orbit, like an electron around a nucleus or planetary bodies in space. It is especially relevant in quantum mechanics and the Bohr model of the atom. The reduced mass of a system accounts for both bodies in a two-body problem, allowing us to simplify the complex interactions to a single body problem.

In the case of an electron revolving around a much heavier proton, the reduced mass \(\mu\) is approximately the mass of the electron because the proton's mass is so much greater (about 1836 times greater). The formula for the reduced mass is \(\mu = \frac{m_e m_p}{m_e + m_p}\), where \(m_e\) is the mass of the electron and \(m_p\) is the mass of the proton. This value is crucial when adjusting the Bohr model for any atom other than hydrogen, as it slightly changes how we calculate the energy levels the electron can occupy. When dealing with our hypothetical particle, with twice the electron's mass, the approximation of \(\mu\) being similar to this new mass leads to modified energy levels and, thus, changes in the transitions and spectral lines.